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The Repulsive Potentials in Alkali Halide Molecules C. Sinistri and C. Margheritis Department of Physical Chemistry, University of Pavia, Italy

Z. Naturforsch. 48a, 987-994 (1993); received May 17, 1993

On the basis of literature values of various spectroscopic quantities, the "experimental" five derivatives (1st to 5th) of the repulsive functions at the equilibrium distance were evaluated for the 20 alkali halide molecules, retaining the truncated Rittner model for the attractive forces. A self-con-sistency test showed that the used experimental values are reliable.

Different analytical forms of the repulsive potential were then critically evaluated by comparison with the experimental derivatives. The repulsive functions were characterized by two, three, four or five empirical parameters. It has been shown that only functions with at least three parameters are sufficiently accurate to reproduce spectroscopic quantities such as ße and ye: the classical two parameter functions appeared too crude in this context.

Key words: Repulsive potentials; Derivatives of the repulsive potentials; Alkali halide molecules.

1. Introduction

None of the proposed potentials for the two body interactions within biatomic molecules seems to sat-isfy a full range of applicability. For this reason the problem was re-examined, taking as a first example the alkali halide molecules. For these compounds, in the gas phase, a large amount of spectroscopic data is available in literature.

The potential energy function <P(r) is usually ex-pressed as the sum of an attractive term A(r) formed by a long range coulombic term and two short range terms connected to the dipolar interaction and to the van der Waals (vdW) forces, respectively, and a repul-sive short range term B(r):

0(r) = A(r) + B(r). (1)

The model normally used is the one originally pro-posed by Rittner [1] and subsequently reviewed by Brumer and Karplus (BK) [2] as "truncated" Rittner (T-Rittner) model:

A(r) = -e2/r - 0.5 (<x+ + a _ ) / r 4 - c + _ / r 6 , (2)

where r is the interionic distance, e the common ab-solute value of the ionic charge, a+, a_ are the ionic polarizabilities and c+_ is the dipole-dipole coeffi-cient.

Reprint requests to Prof. C. Sinistri, Department of Physical Chemistry, Viale Taramelli 16, University of Pavia, 1-27100 Pavia, Italy.

The spectrum of the analytical forms proposed for the repulsive term is particularly wide [3, 4] and in-, volves expressions with two or more parameters. The favorite function appears to be the exponential one, used by Rittner himself:

B(r) = a exp (b r), b< 0 , (3)

although more elaborated functions have been pro-posed, e.g. the one recently given by Ali and Hasan [5]:

B(r) = a r~2 cxp(b r 1 5 ) . (4)

In order to obtain explicit values for the two param-eters a and b, the normal procedure has been to fit the first and second derivatives of (1) at the equilibrium distance re (tf>„ is the n-th derivative):

= 0, *2(re) = ke, (5)

where ke is the force constant that can be calculated from the experimental vibrational frequency coe and the reduced mass p of the molecule.

Values for the higher derivatives of <P(r) at the equi-librium distance can be derived from other spectro-scopic quantities (SQ). The required spectroscopic data and the various repulsive functions to which they are connected are summarized in Table 1.

In order to obtain the values of the repulsive func-tions, one must also evaluate (see (2)) the A(r) term and its derivatives. Thus, apart from the re values which are known with high precision [6], figures for the ionic polarizabilities and the van der Waals coeffi-cients are also necessary. Classical values for these quantities are those by Pauling reported in Table III

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988 C. Sinistri and C. Margheritis • The Repulsive Potentials in Alkali Halide Molecules

Repulsive Connected functions spectroscopic data

D,

b2 K B3 K, «,. Bc

ß 4 ke, oce, Be, o)txe or: ke, oce, Be, ße

B5 oc , Be, cotxt, y B<> = rotational 5 e e e ' e e ' u constants.

Table 1. Connection between spectroscopic quantities and the re-pulsive function at r = re. Z); = dissociation en-ergy in ions, ae = rotation-vibration coupling constant, coe xc = vibrational an-harmonicity constant,

of [2]. More accurate values were reported later [7, 8]. The present calculations are based on this latter set *, but it should be noted that the classical Pauling values yield differences that do not appear substantial.

In addition to (5), the relations between the func-tions <P„ at r = re and the SQ are [3, 10]:

* = - D , , (6)

<P3=-3^2(l + ocecoe/6B2)/re, (7)

<|>4= <P2 coexe(64 n2cn)/h, (8)

<*>4 = 51 <P2/2r2 + 9<P3/re + 1.5<Z>2/<£2

-ßea>l<P2/(4r2eBt), (9)

^s = 2 ye o)£2 <P2/(re ß e ) 3 - 60 <*>2/re3 - 40 <P3/r2

- 1 0 <f2/(re <Z>2) - 10 <*>*/(3 <£2) + 3 *« / r e

+ 1 3 ^ 3 ^ 4 / ( 3 ^ 2 ) , (10)

where c is the velocity of light and h Planck's constant. These equations have been used, as shown later,

to obtain the experimental values of the derivatives Bl + B5. In particular, the data collected by BK [2], supported by [6], were used for the following SQ: coe

(to obtain B2 with (5)), ae and Be (to obtain B3 with (7)), a>exe (to obtain ß 4 with (8)) and ye (to obtain B5

with (10)). On the other hand, these equations can obviously be used also to evaluate the SQ once the analytical expression for B(r) is known. In this way, BK [2] used the exponential function (3), obtained with (5), to calculate the SQ's. They found tha t : " . . . the results for a£ and a>e xc are sat isfactory. . . the calcu-lated values of ye and ße are quite poor, while D{ are

* Polarizabilities from the D D model by BK (see [2], Tables VIII, IX) and vdW coefficients calculated with the Slater-Kirkwood equation [8, 9], A second set of values, based on Shanker's equations and kindly supplied by M. S. Ali (private communication), gives only marginal variations.

almost within the limits of experimental accuracy". They concluded that " . . . the T-Rittner model, like the Rittner model, gives reasonable results for some spectroscopic properties but is quite poor for others. The origin of these discrepancies is not clear, although they must arise from the failure of one or more of the assumptions ...". Similar conclusions were normally reached in other papers along with doubts on the accuracy of the literature values, so that recently the quantities ße and ye were no longer recalculated.

In this paper the problem to obtain reliable repul-sive potentials starting from SQ data was faced focus-ing on the prevision of the derivatives of the repulsive functions and on the significance of the spectroscopic properties that can be obtained.

2. Test of the SQ Data: Limits of Dx, ße, ye

Equations (5) (10) with (1) show that, on the basis of the SQ, immediate figures should be available for B and B 1 + B 5 at the equilibrium distance r e . This is normally so for the derivatives B 1 + B 5 , while diffi-culties arise in the evaluation of the B(re) which is connected to the experimental value D; (see (6)). Ex-perimental D; values for the alkali halides were re-ported in [3 c] and [2] (see Table 2 columns 2 and 3, respectively), the latter data being constantly larger by about 3.6%. According to Gaydon [11] the experi-mental error of these quantities is of about 2.5 kcal/ mole or higher for the fluorides. Moreover, we found that the A(re) and B(re) terms represent, on the aver-age, 85% and 15%, respectively, of the absolute value of 4>(re). Thus, with a fixed value for A(re), an experi-mental uncertainty of 4% in D{, becomes an uncer-tainty of 18-19% in B(re). As a final result, the step D; -* B is accompanied by a factor of about 5 in the percentage error.

Since a direct, precise evaluation of B(re) could not be performed, an indirect way was attempted: the ex-perimental values of Bx (re) + Bs(re) were estimated through (5), (7), (8), (10) and used to obtain the func-tion B(r). The quantities (from (6)) and ße (from (9)) could then be used to test the fitted model.

As regards the quantity ße, which ranges from fairly high absolute values to almost zero (see Table 2), it should be underlined that its weight in the B4(re) term (see (9)), averaged on the 20 alkali halides, is 2.7%, ranging from 0.1% (KBr, RbBr, CsBr) to 8.1% (LiF) (see Table 3). Thus in the step ß 4(r e) ße there is an

989 C. Sinistri and C. Margheritis • The Repulsive Potentials in Alkali Halide Molecules

Table 2. Comparison between the experimental [2, 3 c, 6] and calculated (14) D{ (kcal/mole) and ße ( c m - 1 ) values.

Molecule A . exp ^ i , exp ^ i , calc % Error ße, exp ße, calc

LiF 178.3 184.1 180.6 -1244 .19 + 30 - 1 2 4 4 . 4 0 LiCl 148.0 153.3 148.5 - — 190.13 + 6.6 - 1 9 0 . 4 4 LiBr 142.9 147.8 140.3 - 1 . 8 - - 1 0 8 . 8 5 Lil 134.3 138.7 129.7 - 3 . 4 59.96 N a F 149.0 153.9 152.6 - - 7.004 ± 24.01 - 6 . 7 5 NaCl 127.8 132.6 129.6 - - 8 . 3 4 4 + 1.67 - 8 . 1 6 NaBr 123.6 127.7 124.2 - - 4 . 9 7 4 + 0.93 - 4 . 9 7 Nal 116.6 120.3 115.7 - 0 . 8 - 0 . 4 7 4 + 0.73 - 0 . 5 0 KF 134.2 139.2 139.1 - - 2 . 3 3 4 + 8.339 - 2 . 3 5 KCl 113.2 118.0 116.7 - - 0 . 8 3 4 + 0.1 - 0 . 8 1 KBr 109.4 113.6 109.6 - - 0 . 0 2 + 0.07 - 0 . 0 2 Kl 102.3 106.1 104.0 - 0.043 ± 0.06 0.04 RbF 128.9 133.6 135.5 + 1.4 - 3 . 6 6 9 ± 2.00 - 3 . 8 2 RbCl 108.8 113.4 111.5 - 0.233 + 0.133 0.25 RbBr 105.1 109.0 107.3 - 0 . 0 0 + 0.05 0 . 0 0

Rbl 98.4 101.9 100.5 - 0.0527 ± 0.027 0.05 CsF 125.9 130.5 133.6 + 2.4 3.002 ± 0.66 3.30 CsCl 107.8 112.3 109.7 - 0.5 + 0.067 0.42 CsBr 104.7 108.6 104.4 - 0 . 3 0.002 + 0.0037 0 . 0 0

Csl 97.6 101.1 97.6 - 0.0247 + 0.005 0.03

Table 3. Percentage weight of the terms containing ße in BA exp and ye in B s e x p .

Molecule ße term yt term

LiF 8.1 15.4 LiCl 5.6 13.7 LiBr 5.6 13.2 Lil 5.8 22.8 N a F 0.7 22.1 NaCl 4.3 15.1 NaBr 6.0 13.5 Nal 1.1 15.9 KF 0.7 13.3 KCl 1.5 12.2 KBr 0.1 14.6 Kl 0.4 12.9 RbF 2.1 9.8 RbCl 1.0 11.3 RbBr 0.1 11.7 Rbl 2.5 13.6 CsF 2.5 5.9 CsCl 3.7 9.1 CsBr 0.1 8.8 Csl 2.6 10.2

Table 4. Fitted coefficients a (erg/molecule), b (Ä *), c (Ä 2), d ( k ~ 3 ) , e (Ä"4) for (14).

Mole-cule a b c d e

LiF 0.1190 10" 7 - 7 . 9 3 1 2.525 - 0 . 4 9 9 0.001 LiCl 0.3209 i o -

8 - 3 . 4 8 1 - 0 . 5 3 7 0.465 - 0 . 1 0 1 LiBr 0.3024 i o -

8 - 3 . 1 4 4 - 0 . 5 6 1 0.419 - 0 . 0 8 5 Lil 0.9724 10~ 12 10.434 - 8 . 6 9 0 2.570 - 0 . 2 9 1 N a F 0.1771 10" 10 8.696 - 1 1 . 1 9 3 4.422 - 0 . 6 4 4 NaCl 0.6004 10" 8 - 3 . 1 4 3 - 0 . 7 7 2 0.482 - 0 . 0 8 5 NaBr 0.3424 10" 7 - 5 . 6 9 9 0.879 0.022 - 0 . 0 3 6 Nal 0.1233 10" 9 2.769 - 3 . 5 7 0 1.048 - 0 . 1 1 8 KF 0.1125 i o -

8 - 0 . 3 1 2 - 2 . 8 3 5 1.098 - 0 . 1 5 3 KCl 0.6347 10" 8 - 2 . 6 1 1 - 0 . 7 5 2 0.362 - 0 . 0 5 5 KBr 0.4304 i o -

9 1.277 - 2 . 6 6 8 0.773 - 0 . 0 8 3 Kl 0.3567 i o -

9 1.474 - 2 . 4 6 6 0.669 - 0 . 0 6 9 RbF 0.5558 10" 7 - 7 . 1 5 6 2.019 - 0 . 4 2 4 0.026 RbCl 0.1678 i o -

8 - 0 . 6 8 9 - 1 . 5 4 1 0.471 - 0 . 0 5 2 RbBr 0.2727 i o -

8 - 1 . 0 0 8 - 1 . 4 0 1 0.463 - 0 . 0 5 6 Rbl 0.2207 i o -

10 5.172 - 4 . 1 3 0 0.995 - 0 . 0 9 2 CsF 0.1289 10" 8 - 0 . 8 8 8 - 1 . 6 3 3 0.524 - 0 . 0 6 3 CsCl 0.5100 10" 9 1.100 - 2 . 3 3 5 0.632 - 0 . 0 6 6 CsBr 0.3606 i o -

7 - 4 . 2 3 0 0.312 0.057 - 0 . 0 1 9 Csl 0.1470 10" 9 2.719 - 2 . 7 4 2 0.649 - 0 . 0 5 9

average factor of 30 40 on the percentage error. It is then clear that the ße test is quite strict since it requires a very precise evaluation of J54 along with that of B2

and B3. In order to verify the self-consistency of the SQ a

repulsive function with 5 parameters has been assumed:

B(r) = a exp (b r) exp (c r2) exp (d r3) exp (e r 4 ) , (13)

i.e. ln(B(r)/a) = br + cr2-(- dr3 + er4 . (14)

The five experimental values B1(re)-^-B5(re) can then be used to yield the five constants a + e and thus the function B(r). Table 4 shows the fitted a -=- e coefficients while Table 2 reports the D{ and ße values calculated with (6), (9). The D{ error is taken as zero (dash) when the calculated value is in between the two literature figures: 6 salts fall outside these limits with the largest errors being those of Lil and CsF. The average abso-lute error is 0.5%. The comparison of the ße,cai data with the literature ones [2, 6] is quite significant: ex-cept for CsCl, where the calculated value (0.42) is very close to the lowest experimental limit (0.433), in all other cases the calculated figures are within the exper-imental errors. This can be interpreted as evidence of the reliability of the experimental data and of self-con-sistency of the SQ.

Along with (14), the following function was also studied:

B(r) = a exp(brm) exp(cr"). (15)

This potential gives the same ße values of Table 2 and slightly different D{ ones which anyhow show the same error (0.5%).

In this context, a final remark should be made on the ye quantity, which has been so largely discussed in literature. A detailed analysis of this term shows that

990 C. Sinistri and C. Margheritis • The Repulsive Potentials in Alkali Halide Molecules

its weight in B5(re) is on the average 12.8% (with a minimum of 5.9% in the case of CsF and a maximum of 22.8% for Lil). Thus, on going from ye to B s(re) there is a reduction of the percentage error of about 1/7 and the figures evaluated for B s(r e) should be reliable.

The percentage weights of the terms containing ße

and ye in the experimental ß 4 and B5, respectively, for the 20 alkali halides are reported in Table 3.

3. The Repulsive Potentials

The experimental values B 1 ^ -B 5 , can now be used to test the precision of different analytical forms for the repulsive function B(r) with an increasing number of parameters.

3.1. Fitting Procedure

If one defines Q = BJB, the following relations are immediately obtained:

B,= BQ,

B2=Bi6 + B6i,

B3=B2q+2BiQi+BQ2, (16)

B4=B3q + 3B2Q1 + 3B1Q2+BQ3,

B5= B4Q + 4B3Q1 + 6B2Q2+4B1Q3+ BQ4,

where Q„ is the n-th derivative of Q. It should be observed that the number of Eq. (16)

required is obviously equal to the number of parame-ters in the function B(r) to be fitted and that, once the analytical form of B(r) is assumed, the ^ ( p a r a m -eters) are immediately available. For example, in the case of the simple function (3), only the first two equa-tions (16) are required and Q = b; Q1 = 0. Moreover on

the right-hand side of these equations, the first term is always the most important one, the other ones being successively smaller and normally of only a few per-cent.

The iterative procedure used in the calculations was the following. An initial set of QN was estimated, e.g. starting from (16-2) with Qx = 0, from (16-3) with q2 = 0 and so on. This set was then cycled, taking into account the relations among the Q„, until convergency was reached on the experimental values Bn. The final set gave the explicit parameter values. This fitting pro-cedure has been tested in many cases against the direct solutions which are, normally, hard to obtain.

3.2. The Precision

A reasonable range of the analytical forms of B(r) was analyzed by means of the following 10 types of functions:

a) B(r) with 2 parameters (input: B2, output: B3, B4, B5, B)

This is the easiest and handiest type of repulsive function and thus the most popular in literature. The examined functions are shown in Table 5, i.e. the exponential one (model 1), the gaussian one (model 3), that with r 1 5 (model 5) and finally the function given by Ali and Hasan [5] (model 7). The same Table gives the % errors of the calculated Bn, while Table 6 reports the parameters fitted for these functions. It can be observed that the best results are those given by functions 1 and 7. The comparison betweeen the abso-lute and relative errors shows that these models tend to underestimate the value of the repulsive derivatives. Moreover, the errors in the ß 4 , c a i c and ß 5 , c a i c are generally larger than the weight of the term containing ße in 5 4 exp and ye in ß 5 exp , respectively. This evi-

Model ^ c a l c calc calc ^ 5 , calc

1) a exp (br) 3 . 8 ( 2 . 6 ) 1 . 9 ( -- 1 . 2 ) 6 . 9 ( - 5 . 8 ) 1 6 . 4 ( - 1 0 . 0 )

2 ) arm exp (br) 3 . 8 ( 2 . 9 ) 3 . 1 ( - 2 . 2 ) 1 1 . 2 ( - 3 . 1 )

3 ) a exp (b r2) 3 . 9 ( - 3 . 3 ) 1 8 . 3 ( - 1 8 . 3 ) 5 4 . 8 ( - 5 4 . 8 ) 9 6 . 6 ( - 9 6 . 6 )

4) or™ exp(br 2 ) 3 . 3 ( 2 . 1 ) - 2 . 2 ( - 0 . 2 ) 1 0 . 1 ( 5 . 6 )

5) a exp (br15) 2 . 9 ( -- 0 . 7 ) 9 . 3 ( -- 9 . 3 ) 2 9 . 5 ( - 2 9 . 5 ) 5 4 . 9 ( - 5 4 . 9 )

6 ) arm exp(brus) 3 . 6 ( 0 . 0 ) - 2 . 6 ( - 1 . 2 ) 9 . 8 ( 1 . 2 )

7 ) ar~2 exp(br15) 3 . 5 ( 2 . 0 ) 2 . 0 ( -- 1 . 9 ) 7 . 6 ( - 6 . 7 ) 1 6 . 0 ( - 9 . 4 )

8 ) a exp (br) exp(c r2) 4 . 1 ( 3 . 2 ) - 3 . 2 ( - 2 . 6 ) 1 1 . 1 ( - 4 . 3 )

Table 5. Absolute and rela-tive (in parentheses) mean % errors for the calculated re-pulsive functions 5„ at r = re . For ß c a l c in between the two experimental values (ob-tained from the two Di exp sets of Table 2) the error was taken as zero.

991 C. Sinistri and C. Margheritis • The Repulsive Potentials in Alkali Halide Molecules

Table 6. Fitted coefficients a and b for the two parameter models of Table 5 (r being in Ä, B(r) is in erg/molecule).

Molecule Model 1 Model 3 Model 5 Model 7 Molecule

a - 108 b a •1010 b a - 109 b a - 109 b

LiF 0.1270 - 3 . 7 8 7 0.9271 - 1 . 4 1 5 0.2270 - 2 . 1 8 9 0.1133 - 1 . 3 3 4 LiCl 0.1696 - 3 . 1 6 5 0.9880 - 0 . 9 0 6 0.2604 - 1 . 6 0 0 0.2203 - 1 . 0 2 8 LiBr 0.1762 - 2 . 9 8 6 0.9847 - 0 . 7 9 4 0.2631 - 1 . 4 5 6 0.2574 - 0 . 9 4 3 Lil 0.1946 - 2 . 7 8 0 1.0027 - 0 . 6 6 9 0.2751 - 1 . 2 8 9 0.3284 - 0 . 8 4 8 NaF 0.2509 - 3 . 6 3 7 1.0909 - 1 . 0 7 9 0.3165 - 1 . 8 7 2 0.2473 - 1 . 2 6 7 NaCl 0.3005 - 3 . 0 9 2 1.1326 - 0 . 7 4 5 0.3444 - 1 . 4 3 4 0.4072 - 0 . 9 9 1 NaBr 0.3118 - 2 . 9 4 1 1.1421 - 0 . 6 6 8 0.3506 - 1 . 3 2 4 0.4663 - 0 . 9 1 9 Nal 0.2854 - 2 . 6 8 6 1.0852 - 0 . 5 6 3 0.3290 - 1 . 1 6 2 0.5130 - 0 . 8 0 3 KF 0.3317 - 3 . 3 5 5 1.2589 - 0 . 8 7 9 0.3820 - 1 . 6 2 2 0.3820 - 1 . 1 2 1 KCl 0.4392 - 2 . 9 2 6 1.2970 - 0 . 6 1 9 0.4274 - 1 . 2 7 1 0.6523 - 0 . 9 0 7 KBr 0.3501 - 2 . 6 8 9 1.1498 - 0 . 5 3 9 0.3658 - 1 . 1 3 8 0.6216 - 0 . 8 0 1 Kl 0.4495 - 2 . 6 0 0 1.2520 - 0 . 4 8 0 0.4206 - 1 . 0 5 5 0.8404 - 0 . 7 5 8 RbF 0.3908 - 3 . 2 7 3 1.3818 - 0 . 8 1 8 0.4291 - 1 . 5 4 6 0.4707 - 1 . 0 7 8 RbCl 0.4288 - 2 . 7 8 4 1.2946 - 0 . 5 6 4 0.4235 - 1 . 1 8 3 0.7050 - 0 . 8 4 2 RbBr 0.5481 - 2 . 7 6 1 1.3818 - 0 . 5 2 6 0.4797 - 1 . 1 3 8 0.8985 - 0 . 8 2 7 Rbl 0.5766 - 2 . 5 9 3 1.3785 - 0 . 4 5 8 0.4869 - 1 . 0 2 9 1.0640 - 0 . 7 5 1 CsF 0.3994 - 3 . 1 2 6 1.4833 - 0 . 7 5 7 0.4533 - 1 . 4 5 3 0.5294 - 1 . 0 0 7 CsCl 0.6157 - 2 . 8 0 6 1.5317 - 0 . 5 4 2 0.5340 - 1 . 1 6 5 0.9750 - 0 . 8 4 7 CsBr 0.6646 - 2 . 7 0 5 1.5339 - 0 . 4 9 3 0.5482 - 1 . 0 9 1 1.1218 - 0 . 7 9 9 Csl 0.7124 - 2 . 5 4 9 1.5355 - 0 . 4 3 0 0.5612 - 0 . 9 8 9 1.3409 - 0 . 7 2 9

dence, tested for each salt, prevents the evaluation of ße and ye and thus discourages the use of these func-tions when precision is needed down to this level; in particular it was observed that ße could not be calcu-lated in 74 (and ye in 64) out of the 80 examined cases, while the few favorable ones were always connected with models 1 and 7. It can then be concluded that, in general, values for ße and ye cannot be calculated since the two parameter repulsive potentials appear too crude for the evaluation of these quantities.

Finally, it should be observed that, owing to (16), for each molecule the parameters of Table 6 are con-nected to each other. For example, with the values for a and b of the exponential model 1, the corresponding aG and bG parameters of the gaussian model 3 can be calculated by means of the relations

aG = a &(/> — r e- 1 ) - 1 exp(0.56 r e + 0 . 5 ) ,

bc = 0.5r;1(b-r;1). (17)

Moreover, the following approximate relation can also be applied:

bbG(bU5)~2=lA2, (18)

where b t 5 refers to model 5.

b) B(R) with 3 parameters (input: B L , B 2 , B3;

output: BA,B5,B).

Three of the previously used functions are modified by the introduction of the pre-exponential term rm: models 2, 4 and 6 are thus obtained. For comparison, the function

B(r) = a exp (b r) exp (c r2) (19)

was also taken into account. Tables 5 and 7 summarize the results. It can be ob-

served that the best results are those given by models 4 and 6, where the pre-exponential term yields marked improvements. In this case ße could not be calculated in 43 (and ye in 23) out of 80 cases. In considering these numbers one must refer to Table 3 which, for example, shows that nine molecules require a precision higher than 1.5% in B 4 calc in order to allow the evaluation

® f ße •

Also in this case, the parameters reported in Table 7 are for each molecule interconnected. Useful approxi-mate relations are:

bbG(b1 5)~2 = 1.27 , (20)

( m - m 1 5 ) / ( m 1 5 - mG) = 1.99 ,

where b, m refer to model 2, while subscripts 1.5 and G refer to models 6 and 4, respectively.

992 C. Sinistri and C. Margheritis • The Repulsive Potentials in Alkali Halide Molecules

Table 7. Fitted coefficients a, b and m (or c) for the three parameter models of Table 5 (r being in Ä, B(r) is in erg/molecule).

Molecule Model 2 Model 4 Model 6 Model! 8

a • 108 b m a - 109 b m a -10 9 b m a - 108 b c

LiF 0.0251 - 2 . 0 7 7 - 2 . 2 6 2 0.0491 - 0 . 4 0 0 - 3 . 6 6 1 0.0733 - 0 . 8 0 8 - 3 . 1 9 0 0.3968 - 5 . 5 0 0 0.675 LiCl 0.1203 - 2 . 7 0 8 - 0 . 7 9 7 0.1329 - 0 . 3 8 9 - 3 . 2 1 1 0.2122 - 0 . 9 1 1 - 2 . 4 0 0 0.2501 - 3 . 6 0 7 0.130 LiBr 0.1521 - 2 . 7 6 7 - 0 . 4 1 1 0.1638 - 0 . 3 6 8 - 3 . 0 7 1 0.2565 - 0 . 8 9 6 - 2 . 1 7 8 0.2151 - 3 . 1 9 7 0.058 Lil 0.2034 - 2 . 8 6 4 0.174 0.2155 - 0 . 3 4 3 - 2 . 8 7 8 0.3245 - 0 . 8 8 0 - 1 . 8 5 3 0.1788 - 2 . 7 0 0 - 0 . 0 2 0 NaF 0.0679 - 1 . 9 8 8 - 2 . 7 5 7 0.1341 - 0 . 3 0 2 - 4 . 4 3 6 0.1913 - 0 . 6 8 7 - 3 . 8 7 3 0.9998 - 5 . 2 8 6 0.507 NaCl 0.2515 - 2 . 7 6 3 - 0 . 6 8 3 0.2859 - 0 . 3 3 4 - 3 . 5 9 8 0.4270 - 0 . 8 5 2 - 2 . 6 2 1 0.4194 - 3 . 4 1 2 0.079 NaBr 0.3101 - 2 . 9 2 9 - 0 . 0 2 7 0.3268 - 0 . 3 3 2 - 3 . 3 1 6 0.4801 - 0 . 8 7 5 - 2 . 2 1 4 0.3160 - 2 . 9 5 3 0.003 Nal 0.2870 - 2 . 7 0 4 0.044 0.3933 - 0 . 2 8 3 - 3 . 2 4 4 0.5290 - 0 . 7 7 7 - 2 . 1 4 2 0.2793 - 2 . 6 6 8 - 0 . 0 0 4 KF 0.2011 - 2 . 5 9 5 - 1 . 4 4 7 0.2488 - 0 . 3 4 3 - 3 . 9 5 4 0.3781 - 0 . 8 3 7 - 3 . 1 1 4 0.6759 - 4 . 1 0 0 - 0 . 2 0 0 KCl 0.4282 - 2 . 8 4 4 - 0 . 1 9 4 0.5201 - 0 . 3 0 1 - 3 . 6 1 1 0.7180 - 0 . 8 2 1 - 2 . 4 6 7 0.4828 - 3 . 0 0 6 0.017 KBr 0.3159 - 2 . 1 7 1 - 1 . 2 8 6 0.6676 - 0 . 2 1 9 - 4 . 0 2 5 0.8262 - 0 . 6 1 2 - 3 . 1 0 7 0.6587 - 3 . 1 9 5 0.104 KI 0.4488 - 2 . 7 2 8 0.346 0.7376 - 0 . 2 5 1 - 3 . 4 1 2 0.8858 - 0 . 7 3 5 - 2 . 1 5 4 0.3798 - 2 . 4 7 6 - 0 . 0 2 3 RbF 0.2934 - 2 . 7 9 3 - 0 . 9 6 0 0.3173 - 0 . 3 5 0 - 3 . 7 9 5 0.4862 - 0 . 8 7 8 - 2 . 8 4 5 0.6255 - 3 . 7 4 2 0.120 RbCl 0.3870 - 2 . 3 1 9 - 1 . 1 4 4 0.7265 - 0 . 2 3 6 - 4 . 0 4 2 0.9189 - 0 . 6 5 7 - 3 . 0 7 2 0.7521 - 3 . 2 3 8 0.094 RbBr 0.5526 - 2 . 8 7 3 0.296 0.7730 - 0 . 2 7 3 - 3 . 5 3 6 0.9717 - 0 . 7 8 7 - 2 . 2 5 3 0.4744 - 2 . 6 5 1 - 0 . 0 2 1 Rbl 0.5558 - 2 . 8 5 0 0.729 0.9288 - 0 . 2 5 0 - 3 . 3 8 1 1.0664 - 0 . 7 5 0 - 2 . 0 0 6 0.4042 - 2 . 3 4 4 - 0 . 0 4 5 CsF 0.3575 - 2 . 9 2 4 - 0 . 4 1 7 0.3566 - 0 . 3 5 4 - 3 . 4 8 7 0.5468 - 0 . 9 0 4 - 2 . 4 5 8 0.4894 - 3 . 3 2 2 0.048 CsCl 0.6244 - 2 . 9 4 1 0.347 0.8144 - 0 . 2 8 3 - 3 . 5 2 4 1.0433 - 0 . 8 1 0 - 2 . 2 2 9 0.5197 - 2 . 6 7 6 - 0 . 0 2 6 CsBr 0.6566 - 2 . 9 5 6 0.689 0.9516 - 0 . 2 6 8 - 3 . 4 3 6 1.1442 - 0 . 7 9 1 - 2 . 0 5 6 0.4751 - 2 . 4 6 1 - 0 . 0 4 5 Csl 0.6537 - 2 . 8 5 2 0.900 1.2176 - 0 . 2 3 9 - 3 . 4 0 2 1.3196 - 0 . 7 3 4 - 1 . 9 6 3 0.4593 - 2 . 2 5 5 - 0 . 0 5 0

c) B(r) with 4 parameters (input: Bt -h.B4; output: B5,B)

Two functions of this type were taken into account:

B(r) = a exp (Ib r) exp (c r2) exp (d r3), (21)

B(r) = a rm exp (b r"). (22)

The following results were obtained: B5(re) can be calculated with an absolute error of 3.9 and of 4.6 from (21), (22), respectively. The calculated B(re) shows an error of 3.4 and 3.1, respectively. As expected, the ße

values calculated by these functions are identical to those reported in Table 2, which represent the figures consistent with the fitted experimental coexe. The cal-culated ye show an error which is about 7 times that

Of ^5,calc-

4. Conclusions

The data in Table 5, along with those obtained with the 4-parameter functions, give a quantitative picture of how the precision of the Bn calc increases with the number of fitted parameters in the assumed B(r) func-tion, and also of how, given a repulsive function, the precision of its calculated derivatives decreases as the order of the derivation increases. A comparative analy-sis of the functions with 2 and 3 parameters is reported

in Figs. 1 and 2 for the LiF molecule. As can be ob-served, the 2 parameter functions are rather scattered, while the 3-parameter ones are clustered. This is in agreement with the figures reported in Table 5: the choice of a 2-parameter function is much more critical than that of a multiparameter one.

Moreover it should be observed that an estimate of the errors in the calculated SQ based on the errors in the Bn calc is rather complicated: for example, when ye

is calculated by means of a 2-parameter function, the final expression (see (10)) includes the calculated ß 3 , ß 4 and Bs, each being burdened with its own error. A direct relation is present only between the error of the first of the calculated derivatives and that of its con-nected SQ: e.g. for ae (from B 3 calc) with a 2-parameter function there is a constant factor of 2.2, for a>exe

(from B4 calc) with a 3-parameter function a constant factor of 0.63.

Table 8 summarizes the % absolute errors obtained in the SQ calculated by the models 1 and 7 for the 2-parameter and 4 and 6 for the 3-parameter functions (see Table 5) and (21), (22) in the case of the 4 parame-ters. The ße values in the 2-parameter models are given as "non computable" (NC) since limited, for each equation, to only 3 cases out of 20 on the basis of the previously discussed criteria. In the 3-parame-ter models, for each equation, 10/20 cases could be taken into account: here a zero error was assumed

C. Sinistri and C. Margher i t i s • The Repulsive Potent ia l s in

Fig. 1. Repulsive potentials (erg 1 0 n / m o l e c u l e ) Q f LiF vs. r (Ä) as calculated by the two parameter models. For com-parison, (14) is also reported.

Table 8. Mean absolute % errors in the calculated spectro-scopic quantities. The models are those of Table 5.

SQ 2-parameters 3-parameters 4-parameters

model 1 model 7 model 4 model 6 Eq. (21) Eq.(22)

ae 4.2 4.3 _ _ coe xe 5.9 4.9 3.3 4.1 - -

ß. N C 24.3 * 41.1* 0.0 0.0 ye N C 39.0** 43.1** 27.2 32.9

* Average on 10/20 cases. ** Average on 15/20 cases.

when the ße calc was within the experimental error of ße, exp • Finally, in the 4-parameter models, the ße c a l c

values coincide with the experimental ones (see Table 2). The ye with a 2-parameter model could be obtained, for each equation, in 8/20 cases with an average error of 58.9%; in the 3-parameter model they could be calculated in 15/20 cases.

ii Hal ide Molecules 993

0.6

1 1

model 8

1

LiF

0 .4 -

model 2 -

0 .2 -

eq. 1

model 6

0 . 0 - model 4

i i i i 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4

Fig. 2. Repulsive potentials (erg lO^/molecule) of LiF vs. r (Ä) as calculated by the three parameter models. For com-parison, (14) is also reported.

It can thus be concluded that the analysis of differ-ent analytical expressions for the repulsive potential and mainly that of the significance and limits of the generated SQ has brought light into the problem of the prevision of the ße and ye values. In order to be able to evaluate these quantities, one must deal with functions that are sufficiently precise and however compatible with the weight of the terms containing ße

and ye in the B4 exp and Bs exp of the chosen molecule (see Table 3). Thus, in the case of Na(F, I), K(F, CI, Br, I), Rb (CI, Br) and CsBr molecules, ße can be properly calculated only by a 4-parameter function.

The choice of a proper analytical expression for the repulsive potential B(r) must accomodate the desired precision with the difficulties inherent to the use of a multiparameter function. *

Acknowledgement

This research was partially financed by a CNR grant.

* Fur the r details on the results not repor ted in this pape r can be obta ined f rom the au thors .

994 C. Sinistri and C. Margheritis • The Repulsive Potentials in Alkali Halide Molecules

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